JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Matrices

Let S be the set containing all

3 \times 3

matrices with entries from

{- 1, 0, 1}

. The total number of matrices

A \in S

such that the sum of all the diagonal elements of

A^{T} A

is 6 is ______.

Answer: 5376

Solution

For a

3 \times 3

matrix

A

, the trace of

A^{T} A

equals the sum of squares of all entries of

A

.

If

A = a b c d e f g h i

, then:

Tr (A^{T} A) = a^{2} + b^{2} + c^{2} + d^{2} + e^{2} + f^{2} + g^{2} + h^{2} + i^{2} = 6

Since each entry is from

{- 1, 0, 1}

, each squared entry is either 0 or 1.

We need exactly 6 entries to be non-zero (either

\pm 1

) and 3 entries to be 0.

The number of ways to choose which 6 positions are non-zero:

(6 9) = 84

.

For each such choice, each of the 6 non-zero positions can be either

+ 1

- 1

, giving

2^{6} = 64

possibilities.

Total number of matrices:

(6 9) \times 2^{6} = 84 \times 64 = 5376

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Integer answer